Dolan, one of the most influential Catholic leaders in the U.S. who just wrapped up a term as chairman of the U.S. Conference of Catholic Bishops, lamented some of the elements of President Barack Obama's health care law that caused the Catholic Church to stop short of embracing the law.

"We've been asking for reform in healthcare for a long time. So we were kind of an early supporter in this," Dolan said in an interview with NBC's "Meet the Press" set to air on Sunday. "Where we started bristling and saying, 'Uh-oh, first of all this isn't comprehensive, because it's excluding the undocumented immigrant and it's excluding the unborn baby,' so we began to bristle at that."

In particular, Catholic leaders have expressed concern with Obamacare's requirement that employers and health care providers include access to contraceptive and family-planning services as part of their coverage. Catholic teaching is opposed to abortion rights and the use of contraceptives.

Obama had sought to defuse controversy surrounding the contraceptive mandate in particular by allowing employers to opt against providing coverage for contraceptives, but requiring insurers to communicate the option for such coverage directly to consumers whose employers decline to cover contraceptives because of moral objections.

"So that's when we began to worry and draw back and say, 'Mr. President, please, you're really kind of pushing aside some of your greatest supporters here. We want to be with you, we want to be strong. And if you keep doing this, we're not going to be able to be one of your cheerleaders,'" Dolan explained. "And that sadly is what happened."

Dolan also addressed the topic of same-sex marriage, saying that the debate over marriage rights for gay and lesbian couples is far from settled. The New York cardinal's comments come as Pope Francis has signaled a change in tone toward issues involving gays and lesbians, even if Catholic doctrine (which regards homosexuality as a sin) remains unchanged.

"I think maybe we've been out-marketed, sometimes. We've been caricatured as being anti-gay," Dolan said. "When you have forces like Hollywood, when you have forces like politicians, when you have forces like some opinion-molders that are behind it, it's a tough battle."

But, Dolan cautioned, the Catholic Church is "not going to give up" on the issue of marriage.